Balance Weight, Upthrust and Drag to Find Terminal Speed

For a 0.010 kg mass, a film of thickness 0.30 mm, area 0.10 m² and speed 0.085 m/s this gives about 3.5 x 10⁻³ Pa s. For two plates 1 m² in area, with one moving at 4 m/s against a viscous force of 2 N and η = 0.01 poise (0.001 Pa s), d = η A v / F = 2 x 10⁻³ m, which is 2 mm.

A sphere of radius r moving at speed v in a fluid feels the Stokes drag 6π η r v. At terminal velocity the weight equals the upthrust plus the drag, which gives vₜ = 2 r² (ρ - σ) g / (9 η), where ρ is the density of the sphere and σ that of the fluid. So vₜ depends on r², on the density difference and inversely on η. Two spheres of the same size in the same liquid have terminal speeds in the ratio of (ρ - σ): a gold sphere of density 19.5 and a silver sphere of density 10.5 in a liquid of density 1.5 have speeds in the ratio 18 : 9, so silver goes at half the speed of gold. Moving the same ball from water to glycerine scales the speed by [(ρ - σ2)/(ρ - σ1)] x (η1/η2), which gives about 6.25 x 10⁻⁴ cm/s for a steel ball that falls at 10 cm/s in water.

A ball released from rest accelerates until drag balances its net weight, then moves at constant speed; the speed approaches the terminal value gradually. If a ball enters the liquid from a great height, it arrives faster than the terminal speed, so it decelerates towards the terminal speed instead of speeding up. A small raindrop falling through air likewise reaches a terminal speed that is almost independent of the height it falls from. For a rising air bubble of radius 1 mm moving at 2.1 cm/s in a liquid of density 1.47 g/cc, taking the bubble's own density as negligible, η = 2 r² σ g / (9 v), about 1.5 poise.

For two tubes in series the same flow rate passes through both, so tube 1 of length l and radius r and tube 2 of length 2l and radius 2r have pressure differences in the ratio (l / r⁴) : (2l / 16 r⁴) = 8 : 1.